urgent H1B help

Hi, I am using my OPT which will expire next month. My current boss can not help me get H1B now. So I apply some other places and got three offers form three universities. I want them apply the H1B for me first before I left my current position. My question is whether it is legal to ask all of them file H1B application for me or I only can choose one. Whether different university will affect the application procedure of H1B?
I am really stressed.
Thank you in advance.